1. Field of the Invention
The present invention relates to a method of computer controlled nonlinear optimization which is particularly, but not exclusively, designed for optimum control of production and operating systems described by mathematical models or computer aided design. The present invention also relates to an apparatus for performing such a method.
2. Description of the Prior Art
Optimization techniques are concerned with maximizing or minimizing an objective function which is defined by at least one variable. When the relationship is nonlinear, it is not straightforward to obtain an optimum solution to such a problem. In general, a nonlinear optimization problem is generally expressed by the following equation:
______________________________________ Optimize: f(x) Subject to: gi(x) .gtoreq. 0 ( i .epsilon. MI) hj(x) = 0 ( j .epsilon. ME) ______________________________________
where:
f(x) denotes an objective function;
gi(x) represents non-equality constraints;
hj(x) represents equality constraints;
MI, ME respectively represent character-added sets of non-equality and equality constraints; and
x represents co-ordinates of a point in the n-th space, representing n pieces of variables on the nonlinear optimization problem.
For a nonlinear optimization problem a decisive solution method corresponding to e.g. the simplex method or Karmarkar's method (see U.S. Pat. No. 4,744,028) cannot be obtained. As will be discussed in more detail later, there are many nonlinear optimization methods, such as e.g.: those shown in the book "Nonlinear Programming" (Nikka Giren, 1978) and the book "Iterative Methods of Nonlinear Optimization Problems" by S. L. S. Jacoby, J. S. Kowalik and J. T. Pizzo, published by Prentice Hall in 1972, such as:
(i) the transformation method of optimizing functions incorporating the objective and constraints functions;
(ii) the reduced gradient method in the linear planning expanded to nonlinear optimization;
(iii) sequential quadratic programming for improving the search point by repeating the quadratic programming.
Each of these techniques is generally intended to let the objective function converge at the optimum point that is not ultimately improvable further by updating the search point toward the improvement of the objective function from the starting point of search.